Lecture 28

Today we talked about barriers to proving $P\ne NP$ and other complexity-theory results: specifically, the "relativization" barrier introduced by Baker-Gill-Solovay, and the "natural proofs" barrier introduced by Razborov-Rudich.

We also talked about results that evade them. For example, the Hopcroft-Paul-Valiant Theorem, $TIME\left[f\right(n\left)\right]\ne SPACE\left[f\right(n\left)\right]$, and the Paul-Pippenger-Szemeredi-Trotter Theorem, $DTIME\left[n\right]\ne NTIME\left[n\right]$, do not relativize. Nor do they algebrize, according to the Aaronson-Wigderson notion. (At least, my opinion is that the notion of relativizing/algebrizing makes sense for these theorems, and in this sense they do not relativ/algebrize.)

As for natural proofs, we discussed how proofs by diagonalization do not seem to naturalize, and we mentioned explicit circuit lower bound which are by diagonalization; for example, Kannan's result that ${\Sigma }_4$ not in $SIZE\left(n^{1000}\right)$.

Finally, we mentioned some theorems that neither relativize nor (seem to) naturalize: for example, Vinodchandran's Theorem that $PP$ not in $SIZE\left(n^{1000}\right)$, and Santhanam's improvement of this to an explicit promise problem in $MA$ which isn't in $SIZE\left(n^{1000}\right)$.